Approximating longest cycle in a directed graph

Question

Finding the longest cycle (By cycle I mean cycle with no node repitition) in a directed graph is an NP-hard problem, otherwise we can tell if the graph is Hamiltonian or not. My question is: Is there any alpha-approxiamtion polynomial algorithm for this problem ?

Answer 1

Since the longest directed path problem in a directed graph cannot be approximated in polynomial time within a factor of n^(1-epsilon) for any epsilon > 0 , we can quickly deduce that it is also the case for the longest cycle in a directed graph unless P=NP ( source ).

You can make the reduction as follows:
Choose a vertex v , duplicate v into v1 and v2 , duplicate all concerned arcs as well. Now find the longest directed path from v1 to v2 .
Do that for all vertices in the graph. That gives you the longest directed cycle in the graph.

Conclusion: there is no alpha -approximation in polynomial time for the longest cycle problem in directed graphs (unless P=NP of course).